MIAMI—Florida’s new Republican governor, Ron DeSantis, is making a mark quickly in his first few days in office—moving the state Supreme Court to the right, unveiling a wide-ranging environmental plan and hinting he may take action against the sheriff who oversaw the response to the Parkland school shooting.

Mr. DeSantis, once a little-known congressman, rode to power as a supporter of President Trump and overcame his Democratic opponent in the general election by a wafer-thin margin. He takes the helm of the nation’s largest...